Quartic residues and binary quadratic forms
نویسندگان
چکیده
منابع مشابه
Quartic Residues and Binary Quadratic Forms
Let p ≡ 1 (mod 4) be a prime, m ∈ Z and p m. In this paper we obtain a general criterion for m to be a quartic residue (mod p) in terms of appropriate binary quadratic forms. Let d > 1 be a squarefree integer such that ( d p ) = 1, where ( d p ) is the Legendre symbol, and let εd be the fundamental unit of the quadratic field Q( √ d). Since 1942 many mathematicians tried to characterize those p...
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Let p > 3 be a prime, u, v, d ∈ Z, gcd(u, v) = 1, p u2 − dv2 and (−3d p ) = 1, where ( p ) is the Legendre symbol. In the paper we mainly determine the value of u−v √ d u+v √ d (p−( p3 ))/3 (mod p) by expressing p in terms of appropriate binary quadratic forms. As applications, for p ≡ 1 (mod 3) we obtain a general criterion for m(p−1)/3 (mod p) and a criterion for εd to be a cubic residue of p...
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A reduction theory is developed for binary forms (homogeneous polynomials) of degrees three and four with integer coefficients. The resulting coefficient bounds simplify and improve on those in the literature, particularly in the case of negative discriminant. Applications include systematic enumeration of cubic number fields, and 2-descent on elliptic curves defined over Q. Remarks are given c...
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has only three elements, written h(−23) = 3. There is an binary operation called composition that takes two primitive forms of the same discriminant to a third. Composition is commutative and associative, and makes the set of forms into a group, with identity 〈1, 0,−∆/4〉 for even discriminant and 〈1, 1, (1−∆)/4〉 for odd. From page 49 of Buell [1]: if a form 〈α, β, γ〉 represents a number r primi...
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ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2005
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2004.11.003